aa r X i v : . [ m a t h - ph ] F e b FLAG MANIFOLDS AND GRASSMANNIANS
B. E. EICHINGER
Department of Chemistry, University of Washington, Seattle, Washington 98195-1700
Abstract.
Flag manifolds are shown to describe the relations between configu-rations of distinguished points (topologically equivalent to punctures) embeddedin a general spacetime manifold. Grassmannians are flag manifolds with just twosubsets of points selected out from a set of N points. The geometry of Grassman-nians is determined by a group acting by linear fractional transformations, and theassociated Lie algebra induces transitions between subspaces. Curvature tensorsare derived for a general flag manifold, showing that interactions between a subsetof k points and the remaining N − k points in the configuration is determined bythe coordinates in the flag manifold. Introduction
The symmetric spaces known as Grassmannians have arisen in a wide variety ofphysical contexts in the last few decades. The simplest examples of Grassmanniansare the projective spaces: P R n with Fubini-Study metrics, K¨ahler manifolds, andhyper-K¨ahler manifolds. Higher rank analogs have appeared in the computationof scattering amplitudes,[1] and they are associated with bipartite graphs [2] andquivers.[3] This paper provides an overview of several aspects of these geometricalobjects, highlighting a few features that have interesting physical consequences. Flagmanifolds over quaternions, which will be the primary focus, are related to non-commutative instantons.[4] Configuration Spaces
A configuration space M N , in the sense used here, will be a set of N ≥ N ( x ), where x is a set of coordinates to be determined. The set of points will bedivided into distinguishable subsets by defining a partition of N into two or moreintegers 1 ≤ k µ < N such that Σ µ k µ = N , with k µ being the dimension of the µ -thsubset. To the µ -th subset there is associated a module ψ µ ( x N ), with Ψ N = L µ ψ µ .(The cardinality of the µ -th subset will be understood to be k µ to avoid multiple levels of subscripts in ψ µ .) Each subset, taken individually, is assumed to have boundedmeasure such that the inner product < ψ µ , ψ µ > is finite.A ring acts on Ψ N on the left. The development described here can be done inthe real or complex fields, or the quaternion ring. A symmetry group U ( k µ , K ) overa ring K = R , C , H acting in some representation h µ : ψ µ → h µ ψ µ will preserve theinner product if h µ is in SO ( k µ , R ), SU ( k µ , C ) or U ( k µ , H ) ∼ Sp ( k µ ), depending onthe chosen ring. (To be precise, one should write this as a representation η of h µ acting by ηφ µ . To conserve space and time this will usually be written simply as h µ ψ µ , but unless explicitly stated, this will mean the action of a selected but genericelement of a representation of the group. The number of points in the µ -subset issubsumed in the dimension of the fundamental representation of the group h µ . Ψ N conveys the properties of the physical objects represented as points.) The group h µ is also expected to have an, as yet, unspecified action on coordinates. This is simplydenoted by h µ : ψ µ ( x N ) → h µ ψ µ [ x N ( h µ )] in anticipation of the action of a group on arepresentation. One of our major objectives is to build an explicit set of coordinatesand associated group action. However, we are starting with a primitive notion of afour-dimensional spacetime manifold, and this suggests that the quaternion ring willbe most useful for physical applications. The interested reader will have no troubletranslating the results to the real or complex field.Each subset, also called a system A µ , ≤ µ ≤ m , comes equipped with a symmetrygroup h µ ∼ Sp ( k µ ). The entire set of N points, decomposed into the m subsets, hasa symmetry group acting by(1) H : Ψ N ( · ) → h · h ·· · · ·· · h m ∨ ψ ψ · ψ m . Since H = h × h ×· · ·× h m = N µ h µ and Ψ N is the direct sum of subspaces spannedby the ψ µ as above, this is more simply written as H : Ψ → H Ψ. For now the groupaction on coordinates is acknowledged but not explicitly indicated. The manner inwhich the group acts, as indicated by ∨ , is also not specified.This construction leads to a set of subspaces that are independent of one another.There is nothing in the structure of the H that relates one subspace to another – todo that one needs to enlarge the group for which the off-diagonal elements of(2) G : Ψ N ( · ) → g g · g m g g g ·· · · ·· · g m,m − g mm ∨ ψ ψ · ψ m LAG MANIFOLDS AND GRASSMANNIANS 3 couple the ψ µ subspaces to one another. Since each subspace has a finite innerproduct, so too does Ψ N under the action of a compact group G . The homogeneityof the subspace decomposition, implicit in the notion that N can be partitioned inmany different ways, recommends that G and H belong to the same group, which wenow take to be G ∼ Sp ( N ).[5] Each matrix element in Sp ( N ) is a quaternion. H isa maximal subgroup embedded in G , and this leads to consideration of the naturalprincipal bundle, G ( G/H, H ).[6] The subspaces interact with one another under theaction of Sp ( N ), signifying that the coset G/H induces changes in the ψ µ owingto the presence of other subspaces. The coset space G/H ∼ Sp ( N ) / N Sp ( k µ ) is aflag manifold.[7] A hyper-K¨ahler manifold corresponds to the partition { k = 1 , k = N − } . Grassmannians
The simplest examples of flag manifolds are those for which the N -point configu-ration space is divided into just two subspaces. In our case the associated symmetrygroup H is the product of just two subgroups, H = h k × h n = Sp ( k ) × Sp ( n ) with k + n = N and k ≤ n . The coset space Sp ( k + n ) /Sp ( k ) × Sp ( n ) ∼ G/H is aGrassmann manifold. Note that the notation has been changed so that the sub-script j on h j is the dimension of the fundamental representation of the group. Inthe Grassmannian context this notation is more convenient than that used for thegeneral m -component flag manifold.The Lie algebra sp ( j ) of the group Sp ( j ) consists of skew-symmetric matrices overthe quaternion ring: x ∗ = − x ; x ∈ sp ( j ), where x ∗ is the conjugate transpose of x .[5]The real dimension of Sp ( j ) is j (2 j + 1). The algebra h k ⊕ h n ∈ sp ( k ) ⊕ sp ( n ), with h ∗ j = − h j , consists of diagonal blocks sitting inside x ∈ sp ( n + k ): x = (cid:20) h k p − p ∗ h n (cid:21) , where p is a k × n matrix with quaternion entries. This gives a parameterization of x ∈ Sp ( k + n ) as x = exp (cid:20) p − p ∗ (cid:21) exp (cid:20) h k h n (cid:21) = yh where the coset y ∼ G/H comprises the Grassmann manifold: it is a cross-sectionof the bundle with fiber H. The elements of the coset space y couple the individualcomponents of ψ k to those of ψ n , which can be encoded in a bipartite graph.[2]A coordinate version of the Grassmannian will be helpful to expose several aspectsof the geometry. The Stiefel manifold X = [ X k , X n ] ⊂ g comprises, say, the first k rows of a matrix g ∈ Sp ( k + n ). Here X k is a k × k matrix and X n is a k × n matrix B. E. EICHINGER such that XX ∗ = 1. (Here and throughout the dimension of the unit matrix 1 willbe understood from the context.) The X k part of X can be factored: XX ∗ = X k X ∗ k + X n X ∗ n = X k (1 + Y Y ∗ ) X ∗ k = 1with Y = X − k X n , from which it follows that 1 + Y Y ∗ = ( X ∗ k X k ) − ≥
1. Theprojective space Y is the Grassmann manifold. The real dimension of X is 4 k ( k + n ) − k − k ( k − / kn + 2 k + k . the dimension of Y is 4 kn , and the remainingpart, X ∗ k X k has real dimension k (2 k + 1).Since Xg , g ∈ Sp ( k + n ), is also a subset of Sp ( k + n ), it follows that g acts on X by g : X → ˆ X = [ X k , X n ] (cid:20) A ∗ − C ∗ − B ∗ D ∗ (cid:21) = [ X k A ∗ − X n B ∗ , − X k C ∗ + X n D ∗ ] , where the partitioning of g is compatible with that of X . (The reason for the peculiarlabeling of matrix elements will become apparent shortly.) The transformation X → ˆ X acts by a linear fractional transformation on the Grassmannian, sending Y → ˆ Y ,such that g : Y → ˆ Y = ( A ∗ − Y B ∗ ) − ( − C ∗ + Y D ∗ ) . Given that g g ∗ = 1, it follows that(3) ˆ Y = ( A ∗ − Y B ∗ ) − ( − C ∗ + Y D ∗ ) = ( AY + B )( CY + D ) − . The right hand version of this equation is canonical in the literature,[8] which justifiesthe choice made for the matrix elements in g . Linear fractional transformations arecomposed of rotations, translations, and inversions.Now fix g and vary Y ; using eq. (3) and the unitarity (over H ) of the group it isnot difficult to show that d ˆ Y =( A ∗ − Y B ∗ ) − dY ( CY + D ) − , Y ˆ Y ∗ =( A ∗ − Y B ∗ ) − (1 + Y Y ∗ )( A − BY ∗ ) − , and1 + ˆ Y ∗ ˆ Y =( Y ∗ C ∗ + D ∗ ) − (1 + Y ∗ Y )( CY + D ) − . These pieces are assembled to give the invariant metric ds = Tr[(1 + Y Y ∗ ) − dY (1 + Y ∗ Y ) − dY ∗ ] . In eq. (3) one sees that Y = 0 is mapped to ˆ Y = BD − = − ( A ∗ ) − C ∗ , whichprovides an alternative and very useful representation of the Grassmannian. Thisidentification gives a mixed expression for the metric: ds = Tr[( AA ∗ ) dY ( DD ∗ ) dY ∗ ] = Tr[( A ∗ dY D )( A ∗ dY D ) ∗ ] . LAG MANIFOLDS AND GRASSMANNIANS 5
However, dY = ( dB − Y dD ) D − = [ dB + ( A ∗ ) − C ∗ dD ] D − = ( A ∗ ) − ( A ∗ dB + C ∗ dD ) D − . The last version expresses the one-form dY in terms of the off-diagonalelement of the connection form ω = g ∗ dg = (cid:20) A ∗ − C ∗ − B ∗ D ∗ (cid:21) (cid:20) dA − dB − dC dD (cid:21) = (cid:20) A ∗ dA + C ∗ dC − A ∗ dB − C ∗ dD − B ∗ dA − D ∗ dC B ∗ dB + D ∗ dD (cid:21) = (cid:20) ω ω − ω ∗ ω (cid:21) , giving(4) A ∗ dY D = − ω and ds = Tr( ω ω ∗ )for the coordinate-free version of the metric.The connection form ω is conveniently constructed on the cotangent bundle. Thecotangent space has a basis e = ( e , e , · · · , e N ) on which G acts to the right. Let e denote the basis at the identity of G , corresponding to the point p ∈ x . Thebasis e ( p ) at an arbitrary point p can be pulled back to the identity by the actionof g − ∈ G . That is, e ( p ) = e g . The change in the basis in an infinitesmalneighborhood of p is d e = e dg = e g − dg , and for our group this is[9] d e = e g ∗ dg = e ω, showing that ω is the connection form on the cotangent bundle of the configurationspace M N .A k × n Grassmannian connects the points (vertices or nodes) in two differentsubsets, corresponding to a bipartite graph of k white nodes and n black nodes, withall possible connections constituting the 4 kn real dimensions of the quaternionicGrassmannian. (The connections within the µ -th subset are contained in h µ in aparticular way, as will be seen later.) Within a given bipartite graph one may selectsubsets of vertices from both the white and black nodes, which implies that Schubertvarieties will be of interest. The corresponding graphs will have external edges whichconnect the selected nodes to those remaining in the ( k + n )-vertex set. However, thefull ( k + n )-particle system has no connections to anything else by construction. Tomake those connections the system must be expanded to include a third subspace,thought of as the surroundings. (Connections to the surroundings will require a non-trivial but practical truncation scheme for calculations.) The next section providesone aspect of the connections between subsets. B. E. EICHINGER
Curvature Tensors on Flag Manifolds.
The exterior derivative of ω is dω = d ( g ∗ dg ) = dg ∗ ∧ dg = − g ∗ dgg ∗ ∧ dg = − g ∗ dg ∧ g ∗ dg = − ω ∧ ω , giving(5) dω + ω ∧ ω = 0 . This is the second Maurer-Cartan equation (MC2).[10] It implies that the affinegroup acts on the tangent space of the configuration space, i.e. , the tangent andcotangent bundles are horizontal.[6]The Maurer-Cartan two-form fits into the setting of the flag manifold as follows.Corresponding to the { k µ } partition of N , denote the blocks of g ∈ Sp ( N ) by g αβ .Similarly, ω = ( ω αβ ) , ≤ α, β ≤ m is the matrix written in block form. With thispartitioning in mind, eq. (5) becomes dω µν + Σ mα =1 ω µα ∧ ω αν = 0and in particular, a block on the diagonal is(6) dω µµ + ω µµ ∧ ω µµ + Σ α = µ ω µα ∧ ω αµ = 0 . (Note that the summation convention is not used.)Now suppose that g → H . The corresponding connection form reduces: ω → ¯ ω = ¯ ω × ¯ ω × · · · × ¯ ω mm . All off-diagonal blocks of ω vanish, and one is leftwith d ¯ ω µµ + ¯ ω µµ ∧ ¯ ω µµ = 0 for all µ . The tangent bundles for the m -subspacesare all horizontal, just as is tangent bundle for M N . Thus the tangent bundlesare independent of one another – all subspaces with vanishing MC2 equations areindependent of the other subspaces.However, in a general flag manifold the off-diagonal elements do not vanish, andwhen dω + ω ∧ ω = 0, Cartan[10] identifies the obstruction as the curvature two-formΩ, i.e. , dω + ω ∧ ω = Ω. The diagonal part, dω µµ + ω µµ ∧ ω µµ in eq. (6), is horizontaland the off-diagonal part, which couples the basis vectors that are orthogonal to the k µ -subspace, is vertical.[6] It follows from eq. (6) that a curvature two-form,[11](7) Ω µ = − Σ α = µ ω µα ∧ ω αµ = Σ α = µ ω µα ∧ ω ∗ µα , is associated to every subspace in an irreducible representation of Sp ( N ). The skew-symmetry of ω was used to get the second equality. The cuvature two-forms areclearly non-negative definite. Chern classes may be constructed from the two-forms:( ∧ Ω µ ) ℓ = tr(Ω µ ∧ Ω µ ∧ · · · ∧ Ω µ ) , ℓ terms . For the given k µ -partitioning, consider a change of basis ˆ e = e h , which correspondsto a different selection of cross-section of G/H ( h ⊂ H ).[9] The associated connectionform is defined by d ˆ e = ˆ e ˆ ω . The exterior derivative of ˆ e is d ˆ e = d e h + e dh = e ωh + e dh = e h ˆ ω = ˆ e ˆ ω LAG MANIFOLDS AND GRASSMANNIANS 7 giving h ˆ ω = ωh + dh . The exterior derivative of this equation givesˆΩ = d ˆ ω + ˆ ω ∧ ˆ ω = h ∗ ( dω + ω ∧ ω ) h = h ∗ Ω h demonstrating the tensor character of Ω. This can also be an aid in simplifying theΩ µµ = Σ ν = µ ω µν ∧ ω ∗ µν forms.The off-diagonal components of the connection form are also of interest, and willhelp to shed light on the interpretation of the curvature two-forms. To make thispoint it is convenient to consider just two subspaces, i.e. , the Grassmann case. Theelements of the exterior derivative of the ω block in eq. (5) is dω = − ω ∧ ω − ω ∧ ω , The exterior derivative of this equation givesΩ ∧ ω = ω ∧ Ω which is equivalent to vanishing torsion. It also serves to show that the magnitudes ofthe two curvature tensors are equal in their projections onto the vertical componentof the connection form.Continuing with the case of two subspaces, it follows that from eq. (4) thatΩ = A ∗ dY D ∧ D ∗ dY ∗ A = A ∗ dY (1 + Y ∗ Y ) − ∧ dY ∗ A Ω = D ∗ dY ∗ A ∧ A ∗ dY D = D ∗ dY ∗ (1 + Y Y ∗ ) − ∧ dY D. for which the obvious traces are R =tr[(1 + Y Y ∗ ) − dY (1 + Y ∗ Y ) − ∧ dY ∗ ] R =tr[(1 + Y ∗ Y ) − dY ∗ (1 + Y Y ∗ ) − ∧ dY ] . These are respectively the Ricci two-forms associated with the two subspaces. Thecomponents of the Ricci two-forms are those of the metric tensor, which identifiesthe Grassmannian as an Einstein space.[12]The physical implication of all this is that each subspace has an associated curva-ture two-form that is determined by the interactions with all of the other subspaces.Where M N has been subdivided into just two subspaces, which might be thought ofas a system and its surroundings, the magnitudes (eigenvalues) of the two-forms areequal, suggestive of Newton’s Third Law. Yang-Mills Geometry
The curvature forms specialize to the case of just two points, for which ω is asingle quaternion: Y → q . The curvature two-forms may be rotated by elements of Sp (1) bringing A and D to forms that commute with the identity. This yieldsΩ = (1 + q ¯ q ) − dq ∧ d ¯ q and Ω = ¯Ω = (1 + q ¯ q ) − d ¯ q ∧ dq. B. E. EICHINGER
These are the curvature two-forms for the original Yang-Mills (YM) theory.[13, 14, 15]This demonstrates that Sp (2) /Sp (1) × Sp (1) is the underlying YM geometry. AsAtiyah[15] points out, dq ∧ d ¯ q is self-dual and d ¯ q ∧ dq is anti-self-dual. One goes intothe other by reflection of the 3-space (conjugation).Using well-known group isomorphisms, Sp (2) /Sp (1) × Sp (1) ∼ SO (5) /SO (3) × SO (4) ∼ SO (5) /SO (4) ∼ S ,[5, 16] the original Yang-Mills construction yields a4-sphere geometry. This suggests that one interpret the two Sp (1) factors as thesymmetry groups of two spins, located at antipodes of the sphere. The only availablecoordinates in this construction are those defining the sphere, i.e. , the coordinatesof the Grassmannian. The general flag manifold Sp ( N ) /Sp (1) N constitutes a many-body Yang-Mills theory. The symmetry group of each of the ψ µ ( k µ = 1) is Sp (1),corresponding to the spin degree of freedom of the fiber sitting on each function. Lie Algebra
The infinitesmal generators of the Lie algebra for the symplectic group are builtfrom the Grassmann coordinates, and this is best done in the M (2 , C ) basis forquaternions. Furthermore, it is convenient to use different symbols to label rowand column indices of the H -valued k × n Grassmannian. Define Q = ( ζ αa ); 1 ≤ α ≤ k, ≤ a ≤ n to be the Grassmannian matrix of quaternions in the M (2 , C )representation, with(8) (cid:20) ζ µ − , t − ζ µ − , t ζ µ, t − ζ µ, t (cid:21) = (cid:20) z (1) z (2) − ¯ z (2) ¯ z (1) (cid:21) µt = q µt ; 1 ≤ µ ≤ k, ≤ t ≤ n. In this representation, z (1) µt = x ,µt + ix ,µt and z (2) µt = x ,µt + ix ,µt , with ¯ z being thecomplex conjugate of z . The differential operator ∂/∂ζ αa is now defined such that ∂ζ αa /∂ζ βb = ∂ βb ζ αa = δ αβ δ ab . As an operator in SU (2) this is ∂ µt = (cid:20) ∂/∂z (1) ∂/∂z (2) − ∂/∂ ¯ z (2) ∂/∂ ¯ z (1) (cid:21) µt . There is an additional ‘almost complex’ structure in the symplectic group that ishelpful in computations. One of the su (2) basis vectors is j = (cid:20) − (cid:21) , and its action on a quaternion is complex conjugation: j ′ qj = (cid:20) −
11 0 (cid:21) (cid:20) z z − ¯ z ¯ z (cid:21) (cid:20) − (cid:21) = (cid:20) ¯ z ¯ z − z z (cid:21) = ¯ q. LAG MANIFOLDS AND GRASSMANNIANS 9
The quaternion conjugate to q is q ∗ = j ′ q ′ j in the M (2 , C ) basis. The skew-symmetryof the Lie algebra of Sp ( k + n ) /Sp ( k ) × Sp ( n ) requires elements − w ∗ conjugate to w , and these comprise the matrix − Q ∗ = − ¯ Q ′ .The utility of the preceding representation of conjugation is that it facilitatesdifferentiation of conjugate quaternions (the summation convention is now used): ∂ ¯ ζ αa /∂ζ βb = ∂ βb ¯ ζ αa = ∂ βb ( J ′ αγ ζ γc J ca ) = J ′ αβ J ′ ab = J βα J ba . In the following the short-hand notation J ′ QJ → ¯ Q will be used: it is understoodthat the pre- and post- J factors are of the form 1 ⊗ j (direct product) with appropriatedimension of the unit matrix, 1, to be compatible with the 2 k × n matrix Q . Nowit is seen why use of two different symbols for row and column are useful – it keepsthe pre- and post-multiplicative J factors straight.The infinitesimal generators of the Lie algebra of Sp ( n + k ) are parameterizedby the coordinates of the Grassmannian, stated here without proof (the summationconvention is used, and note also that to avoid a plethora of subscripts, h and H correspond to h and h , respectively): h αβ = ζ αb ∂ βb − ¯ ζ βb ¯ ∂ αb H ab = ζ µa ∂ µb − ¯ ζ µb ¯ ∂ µa p αa = ¯ ∂ αa + ζ αb ζ µa ∂ µb = ( δ αβ + ζ αb ¯ ζ βb ) ¯ ∂ βa + ζ αb H ab = ( δ ab + ζ µa ¯ ζ µb ) ¯ ∂ αb + ζ µa h αµ . It is easy to see that h ∗ = − h, h ∼ sp ( k ) and H ∗ = − H, H ∼ sp ( n ). Furthermore, ¯ p differs from p by quaternion conjugation as shown above. The infinitesimal generatorsare written more succinctly as h = Q∂ ′ − ( Q∂ ′ ) ∗ H = Q ′ ∂ − ( Q ′ ∂ ) ∗ p =(1 + QQ ∗ ) ¯ ∂ + QH ′ . There are many different ways of writing these equations. Let r αβ = ζ αa ∂ βa , sothat h = r − r ∗ = r − ¯ r ′ = r − J ′ r ′ J . It follows that hJ = rJ + ( rJ ) ′ , which isclearly symmetrical: in components ( hJ ) αβ = ( rJ ) αβ + ( rJ ) βα . By encapsulating thegenerators in these symmetrical forms the commutators are more symmetrical than they would be otherwise. They satisfy the following commutation relations:[( hJ ) αβ , ( hJ ) µν ] = − J αµ ( hJ ) βν − J αν ( hJ ) βµ − J βµ ( hJ ) αν − J βν ( hJ ) αµ [( HJ ) ab , ( HJ ) cd ] = − J ac ( HJ ) bd − J ad ( HJ ) bc − J bc ( HJ ) ad − J bd ( HJ ) ac [ h αβ , H ab ] = 0 † [( hJ ) µν , p αa ] = J αµ p νa + J αν p µa ‡ [( HJ ) bc , p αa ] = J ab p αc + J ac p αb [ p αa , p βb ] = − J ab ( hJ ) αβ − J αβ ( HJ ) ab [¯ p αa , p βb ] = δ αβ H ba + δ ab h βα The last two of these equations show that p generates the entire sp ( k + n ) Lie algebraas it must.The upshot of the commutation relations is the following. From eqs. ( † ) and ( ‡ )it follows that [ h µν , p αa ] = δ αν p µa + J αµ ( J p ) νa [ H bc , ¯ p αa ] = − δ ab ¯ p αc + J ac (¯ pJ ) αb . Alternatively, [ h µν , ¯ p αa ] = − δ αµ ¯ p νa − J αν ( J ¯ p ) µa [ H bc , p αa ] = δ ac p αb − J ab ( pJ ) αc , where the versions with ¯ p αa are obtained from eqs. ( † ) and ( ‡ ) by complex conjuga-tion and using the skew-symmetry of h and H .Given two functions, v in the representation space associated with subset A and V in that for subset A , the diagonal elements of the operators act by h µµ v µ ( n µ ) = n µ v µ ( n µ ) and H bb V b ( n b ) = n b V b ( n b ) (the summation convention is suspended in thisparagraph). Then p µb acts on v µ as a raising operator, while ¯ p µb acting on V b is alowering operator. Switching p µb and ¯ p µb gives the opposite action. The presenceof raising and lowering operators is expected, but the important aspect of the p generators is that they act between different subspaces, and in so doing transferexcitations between the two subspaces.The last point to be made here is that for “weak” interactions, weak in the sensethat the nonlinear terms in p αa are negligible, the action of ∂ αa on a quaternionvalued A ∼ ψ µ ( k µ = 1) single particle state can be calculated. Define the H -valuedoperator ∂ αa → ∂ = ∂ + ∂ i + ∂ j + ∂ k = ∂ + ∇ , where ∂ i = ∂/∂x i , such that ∂A = ∂ ( A + A i + A j + A k ) = ( ∂ + ∇ )( A + A )=( ∂ A − ∇ · A ) + ( ∂ A + ∇ A ) + ∇ × A = f − E + B LAG MANIFOLDS AND GRASSMANNIANS 11 where notation has been borrowed from R vector calculus. The standard interpre-tation of electric and magnetic fields as derivatives of the vector potential has beenmade. (The E , B fields have to be handled with standard cartesian coordinates toderive Maxwell’s equations from this point.) In the present setting the identity com-ponent, f , should not be identified with gauge freedom. As the operators ∂ αa act on ψ a ∈ V by P a ∂ αa ψ a they sum the E , B fields, which corresponds to a macroscopicfield acting on a single particle. Conversely, the conjugate acts on a single ψ α ∈ v to react back on V . Recovering something that looks like the electromagnetic fieldis a self-consistency test. The Grassmannian has been claimed to convey interac-tions between subsets, and this derivative recovers one such interaction. But it alsodemonstrates that the electromagnetic vector potential is a term in the representa-tion space Ψ N . The next section will elaborate on other aspects of this changinginterpretation of interactions. Interpretation of Coordinates
Action of G . We began by considering a module over a set of coordinates x , as thesehave now been shown to be all that is required to define both the curvature two-formsof the components of the flag manifold and the infinitesmal generators of the Liealgebra (from the Grassmannian). We did not have to invoke dimensions outside thefour-dimensional spacetime manifold in which our primitive points are embedded toobtain the curvature tensors. We have also seen the left action of g ∈ Sp ( N ) on thesecoordinates: gxH → yH , by linear fractional transformations. In the discussion ofeq. (1) it was pointed out that subsets of points that are independent of one anotherare described by the representation H , but this is nothing other than a reducedrepresentation of G in eq. (2). The same point regarding irreducibility was made inthe discussion of curvature following eq. (6). Systems that interact with one anotherare related by an irreducible representation of G . Owing to the coset structure offlag manifolds, g ∈ G acts on coordinates to the left and the subgroup H acts on theright. The consequences of this for functions now has to be specified.A representation A g of g ∈ G acts by[18] A g Ψ( x ) = Ψ( g − x ) , where Ψ( x ) is now a vector in the representation space of G (the subscript N isimplicit in the context).Geodesics in the group are of the form exp( t g ) , g ∈ sp ( N ), [17, 18] where t ∈ R is a universal time coordinate. The x component of each of the quaternions in theflag manifold is a cyclic time-like variable. One must resist the urge to embed thecoset coordinates in an external Euclidean space. The coordinates define relationsbetween the points in the manifold, and hence relations between the components of the vector bundle Ψ N , independent of any other geometry. A discussion of thehistory of ideas about relational vs. absolute space can be found in ref. [19]. Action of H . Having specified the left action of G , the action of H remains to bequantified. We want to define Ψ N so that it depends only on the coordinates in theflag manifold, and a way to do this is to average over the fiber. A construction frominduced representation theory[20] is appropriate. DefineΨ( x ) = Z σ ( h ) ϕ ( xh ) dh to be this average of ϕ , where dh is the normalized Haar measure on h = N µ h µ and σ ( h ) is a representation of h . The function ϕ ( xh ) is a mapping from the group G ∼ Sp ( N ) to our Hilbert space (with dimension appropriate for the dimension of σ ). The action of η ∈ h on the right of x isΨ( xη ) = Z σ ( h ) ϕ ( xηh ) dh = Z σ ( η − h ) ϕ ( xh ) dh = σ ( η ∗ )Ψ( x )since the normalized Haar measure is invariant to h → ηh . Within the µ -th sub-space the function ψ µ ( x ) lives in a Hilbert space that is invariant to the action of arepresentation σ ( h µ ) : ψ µ ( x ) → σ ( h µ ) ψ µ ( x ).The consequences of the action of H vis-a-vis bipartite graphs is that one seesthe interactions within nodes of a given color only as linear combinations. To seethe interactions amongst the monochromatic nodes in the µ -th subspace in the sameway that one sees the interactions between subspaces as described by the action of G on x , the µ -th subspace has to be decomposed into single particle states, meaningthat Sp ( k ) → Sp ( k ) /Sp (1) k . Quaternions and Special Relativity.
A review of well-known facts on the re-lation between the algebra of quaternions and special relativity is required to setup a mapping between the two. We begin with the two standard representationsof quaternions. In the first, the basis elements ( , i , j , k ) of the quaternion x = { x + x i + x j + x k | x i ∈ R } are interpreted as abstract algebraic objects in thering H that are subject to the usual rules: i = j = k = ijk = − . The alternativerepresentation makes use of the M (2 , C ) basis of matrix elements over 2 × H or M (2 , C ).The product of two quaternions, a, b ∈ H , is(9) ab = ( a b − a · b ) + a b + b a + a × b where a · b = a b + a b + a b and a × b = ( a b − a b ) i +( a b − a b ) j +( a b − a b ) k ,again borrowing symbols from R vector calculus for the dot and cross product. Theconjugate of a quaternion x = x + x is ¯ x = x − x . Using the product in eq. LAG MANIFOLDS AND GRASSMANNIANS 13 (9) it is easy to show that ab = ¯ b ¯ a . Quaternions form a ring, whereas a vector is amodule. One consequence of this is that conjugation by a unit quaternion in H isequivalent to the action of SO (3) as a rotation in R .Now we turn to the other part of the relation in the title of this section. Spe-cial relativity establishes an equivalence class of frames based on the principle thatunit speed (suitably defined) is invariant. Spacetime coordinates, ( ct, x, y, z ) ∼ ( x , x , x , x ), can be used to construct a quaternion, and the obvious mappingtakes ct → x . In support of this assignment one notes that the identity componentof a quaternion commutes with conjugation by a unit quaternion (which rotates thespatial components). To entwine the space and time coordinates the group (actingby conjugation) has to be expanded. This is done by going over to the M (2 , C )representation, and noting that det( q ) = | q | for q ∈ M (2 , C ) is the same as k q k for q ∈ H . The action of L ∈ SL (2 , C ) by L : q → LqL ∗ , where L ∗ is the transposeconjugate of L accomplishes the entwining.The group SL (2 , C ) is a manifold of six real dimensions. Let a matrix in the groupbe parameterized by L = ρ Λ τ , where ρ ∈ SU (2) ∼ S , τ ∈ U (2) /U (1), and the boostis Λ = (cid:20) λ λ − (cid:21) , λ ∈ R + . This parameterization of L is the well-known polar decomposition. A unit quaternion u ∈ SU (2) may be parameterized by u = (cid:20) e iα/ e − iα/ (cid:21) (cid:20) cos( β/
2) sin( β/ − sin( β/
2) cos( β/ (cid:21) (cid:20) e iγ/ e − iγ/ (cid:21) . In constructing the polar decomposition of L it is seen that one of the diag( e iφ/ , e − iφ/ )terms in, say the right ( τ ) side of the diagonal matrix, commutes with the diagonaland is absorbed by ρ . This decomposition accounts for the 3 + 1 + 2 = 6 real dimen-sions of a general matrix in SL (2 , C ). In acting on x , the ρ and τ unitary matricessimply rotate the “vector” part of the quaternion. The Λ piece of L couples the time x ∼ ct coordinate with just one space coordinate since x = Λˆ x Λ := (cid:20) λ λ − (cid:21) (cid:20) ζ ζ − ¯ ζ ¯ ζ (cid:21) (cid:20) λ λ − (cid:21) = (cid:20) λ ζ ζ − ¯ ζ λ − ¯ ζ (cid:21) where ˆ x = τ xτ ∗ . This is the essential reason that special relativity is silent onrotating frames – the boost applies to only one space dimension. (Subsequent ro-tation of the space frame by ρ simply changes its direction.) Velocity v is notonly the tangent space to R , it can also be interpreted as a projective 3-space,the projection being defined by iv = ( ζ − ¯ ζ ) / ( ζ + ¯ ζ ) = ix /x (followed by anarbitrary rotation). This transforms by linear fractional transformations to i ˆ v =( λ ζ − λ − ¯ ζ ) / ( λ ζ + λ − ¯ ζ ) = ( S + ivC ) / ( C + ivS ) in the boosted coordinates ˆ x . Here C = ( λ + λ − ) / S = ( λ − λ − ) / λ is real.The first point to be made about the transformation(10) i ˆ v = ( S + ivC ) / ( C + ivS )is that v = ± i are two fixed points of the transformation. The origin v = 0, ismapped to the imaginary axis v = − iS/C . To make the speed a real quantity,either x or x has to be pure imaginary. The former choice coincides with a Wickrotation. The latter choice is equivalent to switching from the M (2 , C ) basis tothe Pauli basis. The two choices are equivalent modulo i = √−
1. Here the choice x → − ix , iv → v will be made in eq. (10), so that ˆ v = ( S + Cv ) / ( C + Sv ).Now v = 0 is mapped to ˆ v = S/C , resulting in the definition of the boost as( λ − λ − ) / C = cosh(2 ω ) = 1 / √ − ˆ v . The fixed points of the map are now at v = ±
1, signifying that the speed of light is the same in the boosted frame as in theoriginal frame. This is succintly stated: The fixed points at v = ± SL (2 , C ) / ± SL (2 , C ) transformation is that the well-known group isomorphism, SO (1 , ∼ SL (2 , C ) / ±
1, allows one to perform theoperations above in the R vector space with coordinates X = ( ct, x, y, z ), where ˆ L ∈ SO (1 ,
3) acts by the usual matrix multiplication ˆ L : X → X ˆ L . The third observationabout the relativistic tranformation is that the fixed points of the transformation, v = ±
1, translate into a fixed boundary c t − x − y − z = 0 in spacetime. Ofcourse, all this has been well-known for more than a century. A review of these factssets up a particular point of view that is important in the next section. The Projection.
A single term in our flag manifold is a quaternion q µb → x with x ¯ x = k x k . Stated differently, x ¯ x = x + x + x + x = x + xx ′ = b defines a threesphere with radius b >
0. Now consider also the hyperbola y − yy ′ = a . Both y and x are rows consisting of the components of 3-dimensional real vectors. The family ofhyperbolic surfaces, for various values of a , has a boundary at the light cone, a = 0.The hyperbola and sphere both can be projected into B , the 3-dimensional ball, bythe remarkably similar projections(11) y / ( | y | + a ) = x / ( | x | + b ) . On the left(12) [ y / ( | y | + a )][ y / ( | y | + a )] ′ = ( | y | − a ) / ( | y | + a ) = ( | y | − a ) / ( | y | + a ) ≤ . This projects both the y > y = − a ) and y < y = + a )branches into the ball. On the sphere side of eq. (11) we have(13) [ x / ( | x | + b )][ x / ( | x | + b )] ′ = ( b − | x | ) / ( b + | x | ) = ( b − | x | ) / ( b + | x | ) ≤ . LAG MANIFOLDS AND GRASSMANNIANS 15
This is a sterographic projection of the northern/southern hemisphere centered at thesouth/north pole. Both the hyperbola and sphere require two coordinate charts tocover them. The ball B is a velocity space. The single quaternion in the Yang-Millsfield strength is also handled by the projection in eq. (11). However, in the S repre-sentation the x in eq. (13) is four dimensional, which projects to an AdS space. Toremedy this there is a stereographic projection of S into the quaternions[21]. Theseseveral different representations of four-dimensional spacetime geometry clearly re-late to one another through these projections.This projective equivalence shows that one may convert a (spherical) quaternionin the Grassmannian into the corresponding (hyperbolic) “Pauli pseudo-quaternion”,so the two are not so different. However, this only makes sense one term at a time –one must not project the Grassmannian as a whole, as that would defeat the groupstructure. This is not a defect on the Grassmannian side. The Grassmannian isa many-body construction, whereas relativity is strictly valid only as a pair-wiserelation between frames.The light cone boundary, a = 0 in eq. (12), is the boundary of the ball, which isa two sphere. This corresponds to x = 0 in eq. (13). One may interpret a photonas an interaction with vanishing identity component, suggesting that the identitycomponent of a quaternion in the Grassmannian is related to mass. Acknowledgment
The author is grateful for several helpful discussions with Profs. John Sullivanand Gerald Folland.
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